Evaluating the electronic energy differences

P −1

X

k=0

S²_k

σ_k² , σ²_k= τ

4Mpsin²(kπ/P ) (2.199)

For free particles, eq 2.195 becomes

{Sk}⁰= Qk− λ{Sk}

σ²_k + G (2.200)

It’s easy to see that if we choose λk≡ ˜λσ_k², then

e^{−(A0K−AK)}T ({Q_k}, {Qk}⁰)

T ({Qk}, {Qk)} = 1 (2.201)

and the move is always accepted. While this is not true for interacting system, using this kind of λk

greatly enhances the acceptance. The reason is the same as in PIMD: high frequency terms should be uncoupled from the low frequency, physical modes and thus be sampled exactly. As in PIMD, the centroid (k=0 mode) does not move in this scheme since λ_k=0= 0; a different, tunable λ₀is used, depending on the system and on the thermodynamic conditions.

2.6.2.4 Evaluating the electronic energy differences

When computing electronic energy differences E₀^e(R⁰) − E₀^e(R), we want an estimator with an associated error as small as possible, since it is used to evaluate the acceptance in the penalty method. To obtain this, a sampling procedure based on reweighting is employed, similar in spirit

to what happens during the optimization procedure [78]. In this case, we sample N_c electronic configurations according to the probability distribution

P (r|R, R⁰) ∝ |ΨT(r|R)|²+ |ΨT(r|R⁰)|² (2.202) and the two energies E₀^e(R⁰) and E₀^e(R) are obtained as

E₀^e(R) = PN_c

i=1E_L(r_i|R)pi

PN_c i=1pi

, pi= |ΨT(ri|R)|²

P (ri|R, R⁰) (2.203) E^e₀(R⁰) =

PNc

i=1EL(ri|R⁰)p⁰_i PNc

i=1p⁰_i , p⁰_i =|Ψ_T(r_i|R⁰)|²

P (ri|R, R⁰) (2.204)

∆E₀^e =

N_c

X

i=1

E_L(r_i|R⁰) p⁰_i PNc

i=1p⁰_i − E_L(r_i|R) pi

PNc

i=1p_i (2.205)

As a result of the correlation induced by using the same set of electronic configurations, the variance of ∆E^e₀ from eq. 2.205 is smaller than the variance of the two statistically independent energy averages. Moreover, with P (r|R, R⁰) the sampling of the electronic phase space is not limited to configurations that are relevant only for R⁰ or only for R. If, for instance, we sample according to

|ΨT(r_i|R)|², electronic configurations near the nodal surfaces of ΨT(r_i|R) are rarely selected; the same configurations, however, may be relevant for R⁰, since the nodal structure of the wavefunction depends on the nuclear positions. In this case, the reweighting procedure would break since the weights |ΨT(ri|R)|²/ |ΨT(ri|R⁰)|² would be wildly oscillating, as discussed for the optimization.

The problem is avoided with P (r|R, R⁰).

High pressure solid hydrogen

In this chapter we will present the results obtained by DFT-PIMD and CEIMC simulations for high pressure solid hydrogen at low temperatures. In particular, the succession of phase transitions at T=200 K with the recent proposal of yet another solid phase which may be semi-metallic [18]

makes this region of the phase diagram a natural choice for our computations at finite temperature.

Another interesting region involves the boundaries of phase IV at higher temperatures (≈ 400 K), close to the melting line.

In order to assess the stability of one crystal structure with respect to another at finite temperature, one should compute the respective free energies and compare them. Unfortunately, at the moment, the accuracy required to perform a meaningful calculation of the free energies for several candidate structures at finite temperature is beyond our computational resources. However, we will use DFT-PIMD and CEIMC simulations to characterize at a dynamical level relevant crystal structures.

Our main aim is to properly include nuclear quantum effects in our simulations, so that we can directly compare DFT and VMC energetics at finite temperature. In fact, CEIMC simulations will act as a benchmark for the goodness of the exchange-correlation functional employed.

When simulating solids at constant volume, it is crucial to start with the “right” crystal symmetry, since the constraints imposed by the geometry of the cell may not be compatible with the true stable structure: a metastable structure could become de facto stable. Moreover, even if this is not the case, high energy barriers may prevent the transition to the stable phase to occur within the time limit imposed by the simulation. Unfortunately, as explained in chapter 1, experiments cannot provide rigid constraints for the crystal structures.

We will proceed as follows: we will introduce the crystal lattices relevant in the aforementioned regions of the phase diagram, which will be selected as starting configurations in our simulations.

The selection of these lattices will be motivated through a brief overview of past theoretical works performed in similar thermodynamics conditions, with different degrees of approximations. Then, the simulation protocol adopted for both DFT-PIMD and CEIMC simulations will be stated and, finally, our results will be presented and discussed.

44

3.1 Crystal structures

Crystal structures are classified according to their space group, i.e. to the set of symmetry operations which leave the crystal lattice unchanged. In three dimensions, 230 space groups exist and they will be referred using the Hermann-Mauguin notation [101]. As will be explained in the following section, the candidate structures for the different solid phases of hydrogen are based on Pickard and Needs’s works [43, 102]. Several different competitive lattices were found in this search for stable structures for high pressure solid hydrogen: most of them are formed by bidimensional layers stacked in various ways.

Figure 3.1: The C2c structure, based on the lattice proposed at P=300 GPa in the supplementary material of ref. [102]. Right panel: a 3D view, depicting the four layers stacked in an ABCD fashion. The layers are made of molecules nearly parallel to the respective planes. Left panel: a top view of one layer. The molecular centers form a distorted hexagonal lattice. The primitive

cell contains 24 atoms.

For example, a strong candidate for phase III is the C2c structure, depicted in fig. 3.1: four layers, alternating in an ABCDA fashion. The arrangement of the molecules within the layers (see fig. 3.1) creates a non vanishing electric dipole moment, leading to a relatively strong infrared signal [102], compatible at least qualitatively with experimental results [41]. A structure that is competitive at higher pressures (P>250 GPa) is the Cmca12 lattice (fig. 3.2). In this case the molecular layers are arranged in an ABAB fashion: the molecules are completely parallel to the planes.

Figure 3.2: The Cmca12 structure, based on the lattice proposed at P=300 GPa in the supplementary material of ref. [102]. Right panel: a 3D view, depicting the two layers stacked in an AB fashion. The layers are made of molecules that lie parallel to the respective planes. Left panel: a top view of one layer. The arrangement of the molecular centers is similar to the C2c layers, but in this case the distortion from the hexagonal symmetry is larger. The primitive cell

contains 12 atoms.

Another competitive structure is Cmca4 (fig. 3.3): the symmetry group is the same as Cmca12, but in this case the primitive cell contains 4 atoms. It would be more appropriate to talk about this structure in terms of orthorombic symmetry, since distances among in-plane and out-of-plane molecules are comparable; nevertheless, it is still useful when comparing with the other structures.

Figure 3.3: The Cmca4 structure, based on the lattice proposed at P=300 GPa in the supple-mentary material of ref. [102]. Right panel: a 3D view, depicting the layers stacked in an ABAB fashion.Left panel: a top view of one layer. Given the geometry of the cell, talking about layers is

quite arbitrary. The primitive cell contains 4 atoms.